What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure?

To what extent all even dimensional smooth bundles over $S^{2}$ with a symplectic structure are classified? Assume that $E,F$ are two such vector bundles. Does $E\oplus F$ admit a symplectic structure, too?

The question is motivated by existence of a symplectic structure on the the cotangent bundle of manifolds.

What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure?

To what extent all even dimensional smooth bundles over $S^{2}$ with a symplectic structure are classified? Assume that $E,F$ are two such vector bundles. Does $E\oplus F$ or $E\otimes F$ admit a symplectic structure, too?

The question is motivated by existence of a symplectic structure on the the cotangent bundle of manifolds.

What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure?

To what extent all even dimensional smooth bundles over $S^{2}$ with a symplectic structure are classified? Assume that $E,F$ are two such vector bundle, does $E\oplus F$ admit a symplectic structure?

The question is motivated by existence of a symplectic structure on the the cotangent bundle of manifolds.

What is a smooth two dimensional real vector bundle $E$ over $S^{2}$ such that the total space $E$, as a smooth four manifold, does not admit a symplectic structure?

To what extent all even dimensional smooth bundles over $S^{2}$ with a symplectic structure are classified? Assume that $E,F$ are two such vector bundles. Does $E\oplus F$ admit a symplectic structure, too?

The question is motivated by existence of a symplectic structure on the the cotangent bundle of manifolds.